A335385 -id:A335385 - cp's OEIS frontend

A335386 Tri-unitary highly composite numbers: where the number of tri-unitary divisors (A335385) increases to a record.

1, 2, 6, 24, 120, 840, 7560, 83160, 1081080, 18378360, 349188840, 8031343320, 200783583000, 5822723907000, 180504441117000, 6678664321329000, 273825237174489000, 11774485198503027000, 553400804329642269000, 27116639412152471181000, 1437181888844080972593000

Offset: 1

Amiram Eldar, Jun 04 2020

nonn

Amiram Eldar, Table of n, a(n) for n = 1..32

Analogous sequences: A002182 (highly composite), A002110 (unitary), A037992 (infinitary), A293185 (bi-unitary), A318278 (exponential), A306736 (exponential infinitary), A307845 (exponential unitary), A309141 (nonunitary), A322484 (semi-unitary).

Cf. A335385.

f[p_, e_] := If[e == 3 || e == 6, 4, 2]; d[1] = 1; d[n_] := Times @@ (f @@@ FactorInteger[n]); dm = 0; s = {}; Do[If[(d1 = d[n]) > dm, dm = d1; AppendTo[s, n]], {n, 1, 1100000}]; s

A335385(a(n)) = 2^(n-1).

A335387 Tri-unitary harmonic numbers: numbers k such that the harmonic mean of the tri-unitary divisors of k is an integer.

Original entry on oeis.org

1, 6, 45, 60, 90, 270, 420, 630, 2970, 5460, 8190, 9100, 15925, 27300, 36720, 40950, 46494, 47520, 54600, 81900, 95550, 136500, 163800, 172900, 204750, 232470, 245700, 257040, 332640, 409500, 464940, 491400, 646425, 716625, 790398, 791700, 819000, 900900, 929880

Offset: 1

Amiram Eldar, Jun 04 2020

nonn

Equivalently, numbers k such that A324706(k) | (k * A335385(k)).

Differs from A063947 from n >= 18.

			45 is a term since its tri-unitary divisors are {1, 5, 9, 45} and their harmonic mean, 3, in an integer.

Amiram Eldar, Table of n, a(n) for n = 1..200

A324707 is a subsequence.
Analogous sequences: A001599 (harmonic numbers), A006086 (unitary), A063947 (infinitary), A286325 (bi-unitary), A319745 (nonunitary).
Cf. A324706, A335385.

f1[p_, e_] := If[e == 3 || e == 6, 4, 2]; f2[p_, e_] := If[e == 3, (p^4 - 1)/(p - 1), If[e == 6, (p^8 - 1)/(p^2 - 1), p^e + 1]]; f[p_, e_] := p^e * f1[p, e]/f2[p, e]; tuhQ[1] = True; tuhQ[n_] := IntegerQ[Times @@ (f @@@ FactorInteger[n])]; Select[Range[10^4], tuhQ]

cp's OEIS Frontend

A335386 Tri-unitary highly composite numbers: where the number of tri-unitary divisors (A335385) increases to a record.

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